People arrangements in lines

5 Raman and Sanjay are members of a quiz team which has 9 members in total. Two photographs of the quiz team are to be taken. For the first photograph, the 9 members will stand in a line.

1. How many different arrangements of the 9 members are possible in which Raman will be at the centre of the line?
2. How many different arrangements of the 9 members are possible in which Raman and Sanjay are not next to each other?
   For the second photograph, the members will stand in two rows, with 5 in the back row and 4 in the front row.
3. In how many different ways can the 9 members be divided into a group of 5 and a group of 4?
4. For a random division into a group of 5 and a group of 4, find the probability that Raman and Sanjay are in the same group as each other.

6 Four families go to a theme park together. Mr and Mrs Lin take their 2 children. Mr O'Connor takes his 2 children. Mr and Mrs Ahmed take their 3 children. Mrs Burton takes her son. The 14 people all have to go through a turnstile one at a time to enter the theme park.

1. In how many different orders can the 14 people go through the turnstile if each family stays together?
2. In how many different orders can the 8 children and 6 adults go through the turnstile if no two adults go consecutively? Once inside the theme park, the children go on the roller-coaster. Each roller-coaster car holds 3 people.
3. In how many different ways can the 8 children be divided into two groups of 3 and one group of 2 to go on the roller-coaster?

6 Six men and three women are standing in a supermarket queue.

1. How many possible arrangements are there if there are no restrictions on order?
2. How many possible arrangements are there if no two of the women are standing next to each other?
3. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?

2 Find the number of different ways that 6 boys and 4 girls can stand in a line if

1. all 6 boys stand next to each other,
2. no girl stands next to another girl.

5 A company needs to appoint 3 new assistants. 8 candidates are invited for interview; each candidate has a different surname. The candidates are to be interviewed one after another. The personnel officer randomly selects the order in which the candidates are to be interviewed by drawing their names out of a hat. One of the candidates is called Mr Browne and another is called Mrs Green.

1. Calculate the probability that Mr Browne is interviewed first and Mrs Green is interviewed last. 5 of the 8 candidates invited for interview are women and the other 3 are men. The chief executive can't make up his mind who to appoint so he randomly selects 3 candidates by drawing their names out of a hat.
2. Determine the probability that more women than men are selected.